Vtyjkyuk

The Enneper-Weierstrass parametrization is defined in terms of some complex integrals. Certainly if it happens that the function has some singularities, the integral may depend on the path. In Wikipedia, the parametrization is defined to be an integral from the origin to a certain point on the complex plane.

In this post, the accepted answer said that it does not matter where we choose to define the starting point for the parametrization, because two parametrizations will vary up to a constant. I am confused with this statement, in the sense that the "constant" in the statement above is not really a constant, because say if I choose a starting point $z_0$ and another point $z_1$, at each point $z in mathbbC$, the first parametrization at $z$ is defined using an integral along a path $alpha^1_z$ starting from $z_0$, and the second parametrization at $z$ is defined using an integral along a path $alpha_z^2$ stating from $z_1$. The difference of these two integrals will give an integral from $z_0 $ to $z_1$, along the path going along $alpha_z^1$ and then reverse the path $alpha_z^2$. In the post I just quoted, this is the "constant" (as $z$ varies) as mentioned in that post, which I think is no way true, as the path varies with $z$. Can someone explain this to me, since if such problem cannot be solved, then the surface parametrized by choosing different starting point may vary a lot?

Similar situation occurs in choosing different paths to define the parametrization. Say we fix a parametrization and a point $z_0 in mathbbC$. Then we make an another parametrization by redefining the path going to $z_0$, which causes a single point on the surface getting translated. This really makes me confused, since the surface now is deformed so much, which should not be happening.

Remark: I would like to plot the Chen-Gackstatter surface but the above confusion just makes me unable to plot it.